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arXiv:2110.12230

Non-unitary free boson dynamics and the boson sampling problem

Xiao Dong Chen

Published Oct 23, 2021• 0 citations
arXivHTMLPDF

Abstract

We explore the free boson unitary dynamics subject to repeated random forced measurement. The input state is chosen as a Fock state in real space with the particle number conserved in the entire dynamics. We show that each boson is performing a non-unitary quantum walk in real space and its dynamics can be mapped to directed polymers in a random medium with complex amplitude. We numerically show that in the one dimensional system, when the measurement strength is finite, the system is in the frozen phase with all the bosons localized in the real space. Furthermore, these single particle wave functions take the same probability distribution in real space after long time evolution. Due to this property, the boson sampling for the output state becomes easy to solve. We further investigate circuit with non-local unitary dynamics and numerically demonstrate that there could exist a phase transition from a localized phase to a delocalized phase by varying the measurement strength.

DOI
10.48550/arxiv.2110.12230

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We explore the free boson unitary dynamics subject to repeated random forced measurement. The input state is chosen as a Fock state in real space with the particle number conserved in the entire dynamics. We show that each boson is performing a non-unitary quantum walk in real space and its dynamics can be mapped to directed polymers in a random medium with complex amplitude. We numerically show that in the one dimensional system, when the measurement strength is finite, the system is in the frozen phase with all the bosons localized in the real space. Furthermore, these single particle wave functions take the same probability distribution in real space after long time evolution. Due to this property, the boson sampling for the output state becomes easy to solve. We further investigate circuit with non-local unitary dynamics and numerically demonstrate that there could exist a phase transition from a localized phase to a delocalized phase by varying the measurement strength.
Quantum simulation in its current state faces experimental overhead in terms of physical space and cooling. We propose boson sampling as an alternative compact synthetic platform performing at room temperature. Identifying the capability of estimating matrix permanents, we explore the applicability of boson sampling for tackling the dynamics of quantum systems without having access to information about the full state vector. By mapping the time-evolution unitary of a Hamiltonian onto an interferometer via continuous-variable gate decompositions, we present proof-of-principle results of localization characteristics of a single particle. We study the dynamics of one-dimensional tight-binding systems in the clean and quasiperiodic-disordered limits to observe Bloch oscillations and dynamical localization, and the delocalization-to-localization phase transition in the Aubry-Andr\'e-Harper model, respectively. Our computational results obtained using boson sampling are in complete agreement with the dynamical and static results of noninteracting tight-binding systems obtained using conventional numerical calculations. Additionally, our study highlights the role of the number of sampling measurements or ``shots'' for simulation accuracy.
Entanglement phase transitions driven by quantum measurements have emerged as a central paradigm in open quantum many-body physics. Such phase transitions are well established for systems with finite local Hilbert-space dimensions, such as qubits and fermions, while their realization in bosonic systems with unbounded local occupation numbers remains poorly understood. Even in the absence of interactions, number states of bosons are intrinsically non-Gaussian, preventing the use of standard correlation-matrix approaches. To address this problem, we develop a replica-free Keldysh field-theoretic framework that expresses the Renyi entropy of bosonic systems initialized in on-site Fock states in terms of permanents of matrices constructed from single-particle Green's functions. Applying this framework to a continuously monitored one-dimensional cross-stitch lattice conditioned on the no-click trajectory, we uncover a transition from volume-law to logarithmic entanglement scaling. We show that the transition is controlled by a restructuring of the non-Hermitian spectrum that changes the number of long-lived modes from extensive to finite. In the strongly monitored regime, bosons dynamically condense into a microscopic number of slowest-decaying modes, producing logarithmic entanglement scaling, whereas an extensive manifold of long-lived modes at weak monitoring gives rise to volume-law entanglement. Our results establish a distinct mechanism for measurement-induced entanglement transitions in free bosonic systems and provide a computationally efficient diagnostic of the measurement-induced bosonic condensation.
One of the main postulates of quantum mechanics is that measurements destroy quantum coherence (wave function collapse). Recently it was discovered that in a many-body system dilute local measurements still preserve some coherence across the entire system. As the measurement density is increased, a phase transition occurs that is characterized by the disentanglement of different parts of the system. Unfortunately, this transition is impossible to observe experimentally for macroscopic systems because it requires an exponentially costly full tomography of the many-body wave function or a comparison with the simulation on an oracle classical computer. In this work we report the discovery of another measurement-induced phase transition that can be observed experimentally if quantum dynamics can be reversed. On one side of this phase transition the quantum information encoded in some part of the Hilbert space is fully recovered after the time inversion. On the other side, all quantum information is corrupted. This transition also manifests itself as the change in the behavior of the probability to observe the same measurement outcome in the process that consists of identical blocks repeated many times. In each block the unitary evolution is followed by the measurement. On one side of the transition the probability decreases exponentially with the number of repetitions, on the other it tends to a constant as the number of repetitions is increased. We confirm the existence of the proposed phase transition through numerical simulations of realistic quantum circuits and analytical calculations using an effective random-matrix theory model.
We develop a random sampling method for calculating the time evolution of the R\'enyi entanglement entropy after a quantum quench from an insulating state in free boson systems. Because of the non-Gaussian nature of the initial state, calculating the R\'enyi entanglement entropy calls for the exponential cost of computing a matrix permanent. We numerically demonstrate that a simple random sampling method reduces the computational cost of a permanent; for an ${N}_{\mathrm{s}}\ifmmode\times\else\texttimes\fi{}{N}_{\mathrm{s}}$ matrix corresponding to Ns{N}_{\mathrm{s}}Ns​ sites at half filling, the sampling cost becomes $O({2}^{\ensuremath{\alpha}{N}_{\mathrm{s}}})$ with a constant $\ensuremath{\alpha}\ensuremath{\ll}1$, in contrast to the conventional algorithm with the O(2Ns)O({2}^{{N}_{\mathrm{s}}})O(2Ns​) number of summations requiring the exponential time cost. Although the computational cost is still exponential, this improvement allows us to obtain the entanglement entropy dynamics in free boson systems for more than 100 sites. We present several examples of the entanglement entropy dynamics in low-dimensional free boson systems.
Spin chains with quenched disorder exhibit rich critical behavior, often captured by real-space renormalization group (RSRG) techniques. However, the physics of such systems in the presence of random measurements (i.e., non-Hermitian dephasing) remains largely unexplored. The interplay between measurements and unitary dynamics gives rise to novel phases and phase transitions in monitored quantum systems. In this work, we investigate the disordered XX spin chain subject to stochastic local measurements in the XXX and YYY bases. By mapping the monitored chain to a non-Hermitian spin ladder with complex couplings, we propose an RSRG-for-excited-states (RSRG-X) approach for this open-system setting. Our analysis reveals a new class of strongly disordered fixed points that emerge due to non-unitarity, broadening the landscape of critical phenomena accessible via RSRG.
We make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a function of time t due to evolution generated by spatially local quadratic bosonic Hamiltonians. We obtain an upper bound on the scaling of t with the number of bosons n for which approximate sampling is classically efficient. We also obtain a lower bound on the scaling of t with n for which any instance of the boson sampling problem reduces to this problem and hence implies that the problem is hard, assuming the conjectures of Aaronson and Arkhipov [Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (ACM Press, New York, New York, USA, 2011), p. 333]. This establishes a dynamical phase transition in sampling complexity. Further, we show that systems in the Anderson-localized phase are always easy to sample from at arbitrarily long times. We view these results in light of classifying phases of physical systems based on parameters in the Hamiltonian. In doing so, we combine ideas from mathematical physics and computational complexity to gain insight into the behavior of condensed matter, atomic, molecular, and optical systems.